Subgroups — C₂×C₆₇×#3486 #30742033

Order 17152.

11254 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3486	C₂	1	32	yes	maximal, direct product factor
4	C₂×D₄×Q₈×C₆₇	C₂	1	8	yes	maximal, complemented
5	C₂×#548×C₆₇	C₂	1	8	yes	maximal, complemented
6	C₂×#536×C₆₇	C₂	1	6	yes	maximal, complemented
7	C₂×#525×C₆₇	C₂	1	6	yes	maximal, complemented
8	C₂×#519×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×2₋¹⁺⁴×C₆₇	—	2	16	no	—
10	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	16	yes	complemented
11	C₂²×#41×C₆₇	C₂²	1	6	yes	—
12	C₂²×#41×C₆₇	C₂²	1	4	yes	complemented
13	C₂³×Q₈×C₆₇	—	2	16	no	—
14	C₂³×Q₈×C₆₇	C₂²	1	3	yes	—
15	C₂³×Q₈×C₆₇	C₂²	1	8	yes	—
16	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×Q₈×C₆₇	C₂²	1	16	yes	complemented
18	C₂³×Q₈×C₆₇	C₂²	1	4	yes	complemented
19	C₂⁴×C₄×C₆₇	C₂²	1	3	yes	—
20	D₄×Q₈×C₆₇	C₂²	1	128	yes	complemented
21	#548×C₆₇	C₂²	1	128	yes	complemented
22	#536×C₆₇	C₂²	1	96	yes	complemented
23	C₂×C₄⋊Q₈×C₆₇	C₂²	1	12	yes	complemented
24	C₂×#124×C₆₇	C₂²	1	12	yes	complemented
25	#525×C₆₇	C₂²	1	96	yes	complemented
26	C₂×#123×C₆₇	C₂²	1	12	yes	—
27	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	12	yes	—
28	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	24	yes	complemented
29	C₂×#121×C₆₇	C₂²	1	12	yes	—
30	C₂×#120×C₆₇	C₂²	1	4	yes	—
31	#519×C₆₇	C₂²	1	16	yes	complemented
32	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
33	C₂×C₄×D₄×C₆₇	C₂²	1	24	yes	—
34	C₂×#117×C₆₇	C₂²	1	6	yes	—
35	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
36	2₋¹⁺⁴×C₆₇	—	2	128	no	—
37	C₂×#41×C₆₇	—	2	32	no	—
38	C₂×#41×C₆₇	—	2	48	no	—
39	C₂×#41×C₆₇	C₂³	1	48	yes	—
40	C₂×#41×C₆₇	C₂³	1	32	yes	complemented
41	C₂²×Q₈×C₆₇	—	2	64	no	—
42	C₂²×Q₈×C₆₇	—	2	32	no	—
43	C₂²×Q₈×C₆₇	—	2	128	no	—
44	C₂²×Q₈×C₆₇	—	2	24	no	—
45	C₂²×Q₈×C₆₇	D₄	1	16	yes	complemented
46	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
47	C₂²×Q₈×C₆₇	C₂³	1	12	yes	—
48	C₂²×Q₈×C₆₇	C₂³	1	24	yes	—
49	C₂²×Q₈×C₆₇	C₂³	1	64	yes	—
50	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
51	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
52	C₂²×Q₈×C₆₇	C₂³	1	32	yes	complemented
53	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
54	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
55	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
56	C₂³×C₄×C₆₇	—	2	24	no	—
57	C₂³×C₄×C₆₇	C₂³	1	12	yes	—
58	C₂³×C₄×C₆₇	C₂³	1	12	yes	—
59	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
60	C₂³×C₄×C₆₇	C₂³	1	24	yes	—
61	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
62	C₄⋊Q₈×C₆₇	C₂³	1	96	yes	complemented
63	#124×C₆₇	C₂³	1	96	yes	complemented
64	#123×C₆₇	C₂³	1	96	yes	—
65	C₂²⋊Q₈×C₆₇	C₂³	1	96	yes	—
66	C₂²⋊Q₈×C₆₇	C₂³	1	192	yes	complemented
67	#121×C₆₇	C₂³	1	96	yes	—
68	#120×C₆₇	C₂³	1	32	yes	—
69	C₄×Q₈×C₆₇	C₂³	1	64	yes	complemented
70	C₄×D₄×C₆₇	C₂³	1	192	yes	—
71	#117×C₆₇	C₂³	1	48	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	24	yes	—
74	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
75	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
76	C₂×C₂²⋊C₄×C₆₇	C₂³	1	24	yes	—
77	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
78	C₂⁴×C₆₇	—	2	4	no	—
79	C₂⁴×C₆₇	—	2	8	no	—
80	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
81	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
82	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
84	#41×C₆₇	—	2	128	no	—
85	#41×C₆₇	—	2	192	no	—
86	C₂×Q₈×C₆₇	—	2	256	no	—
87	C₂×Q₈×C₆₇	—	2	128	no	—
88	C₂×Q₈×C₆₇	—	2	48	no	—
89	C₂×Q₈×C₆₇	—	2	96	no	—
90	C₂×Q₈×C₆₇	—	2	24	no	—
91	C₂×Q₈×C₆₇	C₂×D₄	1	32	yes	—
92	C₂×Q₈×C₆₇	C₂×D₄	1	64	yes	complemented
93	C₂×Q₈×C₆₇	C₂⁴	1	24	yes	—
94	C₂×Q₈×C₆₇	C₂⁴	1	48	yes	—
95	C₂×Q₈×C₆₇	C₂⁴	1	16	yes	—
96	C₂×D₄×C₆₇	—	2	16	no	—
97	C₂×D₄×C₆₇	—	2	32	no	—
98	C₂×D₄×C₆₇	—	2	24	no	—
99	C₂×D₄×C₆₇	C₂⁴	1	32	yes	—
100	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
101	C₂×D₄×C₆₇	C₂⁴	1	24	yes	—
102	C₂²×C₄×C₆₇	—	2	12	no	—
103	C₂²×C₄×C₆₇	—	2	48	no	—
104	C₂²×C₄×C₆₇	—	2	96	no	—
105	C₂²×C₄×C₆₇	—	2	48	no	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	24	yes	—
107	C₂²×C₄×C₆₇	C₂⁴	1	48	yes	—
108	C₂²×C₄×C₆₇	C₂⁴	1	48	yes	—
109	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
110	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
111	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
113	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
114	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
115	C₄⋊C₄×C₆₇	C₂⁴	1	48	yes	—
116	C₄⋊C₄×C₆₇	C₂⁴	1	96	yes	—
117	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
118	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
119	C₂²⋊C₄×C₆₇	C₂⁴	1	96	yes	—
120	C₄²×C₆₇	C₂⁴	1	48	yes	—
121	C₂³×C₆₇	—	2	32	no	—
122	C₂³×C₆₇	—	2	8	no	—
123	C₂³×C₆₇	—	2	4	no	—
124	C₂³×C₆₇	—	2	4	no	—
125	C₂³×C₆₇	—	2	4	no	—
126	C₂³×C₆₇	—	2	16	no	—
127	C₂³×C₆₇	—	2	8	no	—
128	C₂³×C₆₇	—	2	16	no	—
129	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
130	C₂³×C₆₇	2₋¹⁺⁴	1	8	yes	complemented
131	C₂³×C₆₇	C₂⁵	1	8	yes	—
132	C₂³×C₆₇	C₂⁵	1	2	yes	—
133	C₂³×C₆₇	C₂⁵	1	4	yes	—
134	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
135	Q₈×C₆₇	—	2	96	no	—
136	Q₈×C₆₇	—	2	48	no	—
137	Q₈×C₆₇	C₂²×D₄	1	64	yes	—
138	D₄×C₆₇	—	2	32	no	—
139	D₄×C₆₇	—	2	64	no	—
140	D₄×C₆₇	—	2	48	no	—
141	C₂×C₄×C₆₇	—	2	8	no	—
142	C₂×C₄×C₆₇	—	2	24	no	—
143	C₂×C₄×C₆₇	—	2	12	no	—
144	C₂×C₄×C₆₇	—	2	16	no	—
145	C₂×C₄×C₆₇	—	2	96	no	—
146	C₂×C₄×C₆₇	—	2	96	no	—
147	C₂×C₄×C₆₇	C₂²×D₄	1	24	yes	—
148	C₂×C₄×C₆₇	C₂²×D₄	1	48	yes	—
149	C₂×C₄×C₆₇	C₂⁵	1	16	yes	—
150	C₂×C₄×C₆₇	C₂⁵	1	12	yes	—
151	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
152	C₂×C₄×C₆₇	C₂⁵	1	12	yes	—
153	C₂²×C₆₇	—	2	32	no	—
154	C₂²×C₆₇	—	2	4	no	—
155	C₂²×C₆₇	—	2	4	no	—
156	C₂²×C₆₇	—	2	4	no	—
157	C₂²×C₆₇	—	2	8	no	—
158	C₂²×C₆₇	—	2	16	no	—
159	C₂²×C₆₇	—	2	8	no	—
160	C₂²×C₆₇	—	2	16	no	—
161	C₂²×C₆₇	—	4	16	no	—
162	C₂²×C₆₇	C₂³×D₄	1	4	yes	—
163	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
164	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	4	yes	—
165	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	8	yes	complemented
166	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
167	C₄×C₆₇	—	2	8	no	—
168	C₄×C₆₇	—	2	12	no	—
169	C₄×C₆₇	—	2	16	no	—
170	C₄×C₆₇	C₂³×D₄	1	24	yes	—
171	C₂×#3486	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂×C₆₇	—	2	4	no	—
173	C₂×C₆₇	—	2	8	no	—
174	C₂×C₆₇	—	4	8	no	—
175	C₂×C₆₇	#3486	1	4	yes	direct product factor
176	C₂×C₆₇	C₂⁴×D₄	1	1	yes	characteristic
177	C₂×C₆₇	C₂²×2₋¹⁺⁴	1	2	yes	—
178	C₂²×2₋¹⁺⁴	C₂×C₆₇	1	1	yes	characteristic, complemented
179	C₂⁴×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
180	#3486	C₂×C₆₇	1	32	yes	direct product factor
181	C₂×D₄×Q₈	C₂×C₆₇	1	8	yes	complemented
182	C₂×#548	C₂×C₆₇	1	8	yes	complemented
183	C₂×#536	C₂×C₆₇	1	6	yes	complemented
184	C₂×#525	C₂×C₆₇	1	6	yes	complemented
185	C₂×#519	C₂×C₆₇	1	1	yes	characteristic, complemented
186	C₆₇	C₂×#3486	1	1	yes	characteristic, direct product factor, Sylow
187	C₂×2₋¹⁺⁴	—	2	16	no	—
188	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	16	yes	complemented
189	C₂²×#41	C₂²×C₆₇	1	6	yes	—
190	C₂²×#41	C₂²×C₆₇	1	4	yes	complemented
191	C₂³×Q₈	—	2	16	no	—
192	C₂³×Q₈	C₂²×C₆₇	1	3	yes	—
193	C₂³×Q₈	C₂²×C₆₇	1	8	yes	—
194	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
195	C₂³×Q₈	C₂²×C₆₇	1	16	yes	complemented
196	C₂³×Q₈	C₂²×C₆₇	1	4	yes	complemented
197	C₂⁴×C₄	C₂²×C₆₇	1	3	yes	—
198	D₄×Q₈	C₂²×C₆₇	1	128	yes	complemented
199	#548	C₂²×C₆₇	1	128	yes	complemented
200	#536	C₂²×C₆₇	1	96	yes	complemented
201	C₂×C₄⋊Q₈	C₂²×C₆₇	1	12	yes	complemented
202	C₂×#124	C₂²×C₆₇	1	12	yes	complemented
203	#525	C₂²×C₆₇	1	96	yes	complemented
204	C₂×#123	C₂²×C₆₇	1	12	yes	—
205	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	12	yes	—
206	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	24	yes	complemented
207	C₂×#121	C₂²×C₆₇	1	12	yes	—
208	C₂×#120	C₂²×C₆₇	1	4	yes	—
209	#519	C₂²×C₆₇	1	16	yes	complemented
210	C₂×C₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
211	C₂×C₄×D₄	C₂²×C₆₇	1	24	yes	—
212	C₂×#117	C₂²×C₆₇	1	6	yes	—
213	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
214	2₋¹⁺⁴	—	2	128	no	—
215	C₂×#41	—	2	32	no	—
216	C₂×#41	—	2	48	no	—
217	C₂×#41	C₂³×C₆₇	1	48	yes	—
218	C₂×#41	C₂³×C₆₇	1	32	yes	complemented
219	C₂²×Q₈	—	2	64	no	—
220	C₂²×Q₈	—	2	32	no	—
221	C₂²×Q₈	—	2	128	no	—
222	C₂²×Q₈	—	2	24	no	—
223	C₂²×Q₈	D₄×C₆₇	1	16	yes	complemented
224	C₂²×Q₈	C₂³×C₆₇	1	6	yes	—
225	C₂²×Q₈	C₂³×C₆₇	1	12	yes	—
226	C₂²×Q₈	C₂³×C₆₇	1	24	yes	—
227	C₂²×Q₈	C₂³×C₆₇	1	64	yes	—
228	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
229	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
230	C₂²×Q₈	C₂³×C₆₇	1	32	yes	complemented
231	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
232	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
233	C₂²×D₄	C₂³×C₆₇	1	6	yes	—
234	C₂³×C₄	—	2	24	no	—
235	C₂³×C₄	C₂³×C₆₇	1	12	yes	—
236	C₂³×C₄	C₂³×C₆₇	1	12	yes	—
237	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
238	C₂³×C₄	C₂³×C₆₇	1	24	yes	—
239	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
240	C₄⋊Q₈	C₂³×C₆₇	1	96	yes	complemented
241	#124	C₂³×C₆₇	1	96	yes	complemented
242	#123	C₂³×C₆₇	1	96	yes	—
243	C₂²⋊Q₈	C₂³×C₆₇	1	96	yes	—
244	C₂²⋊Q₈	C₂³×C₆₇	1	192	yes	complemented
245	#121	C₂³×C₆₇	1	96	yes	—
246	#120	C₂³×C₆₇	1	32	yes	—
247	C₄×Q₈	C₂³×C₆₇	1	64	yes	complemented
248	C₄×D₄	C₂³×C₆₇	1	192	yes	—
249	#117	C₂³×C₆₇	1	48	yes	—
250	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
251	C₂×C₄⋊C₄	C₂³×C₆₇	1	24	yes	—
252	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
253	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
254	C₂×C₂²⋊C₄	C₂³×C₆₇	1	24	yes	—
255	C₂×C₄²	C₂³×C₆₇	1	12	yes	—
256	C₂⁴	—	2	4	no	—
257	C₂⁴	—	2	8	no	—
258	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
259	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
260	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
261	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
262	#41	—	2	128	no	—
263	#41	—	2	192	no	—
264	C₂×Q₈	—	2	256	no	—
265	C₂×Q₈	—	2	128	no	—
266	C₂×Q₈	—	2	48	no	—
267	C₂×Q₈	—	2	96	no	—
268	C₂×Q₈	—	2	24	no	—
269	C₂×Q₈	C₂×D₄×C₆₇	1	32	yes	—
270	C₂×Q₈	C₂×D₄×C₆₇	1	64	yes	complemented
271	C₂×Q₈	C₂⁴×C₆₇	1	24	yes	—
272	C₂×Q₈	C₂⁴×C₆₇	1	48	yes	—
273	C₂×Q₈	C₂⁴×C₆₇	1	16	yes	—
274	C₂×D₄	—	2	16	no	—
275	C₂×D₄	—	2	32	no	—
276	C₂×D₄	—	2	24	no	—
277	C₂×D₄	C₂⁴×C₆₇	1	32	yes	—
278	C₂×D₄	C₂⁴×C₆₇	1	16	yes	—
279	C₂×D₄	C₂⁴×C₆₇	1	24	yes	—
280	C₂²×C₄	—	2	12	no	—
281	C₂²×C₄	—	2	48	no	—
282	C₂²×C₄	—	2	96	no	—
283	C₂²×C₄	—	2	48	no	—
284	C₂²×C₄	C₂×D₄×C₆₇	1	24	yes	—
285	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
286	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
287	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
288	C₂²×C₄	C₂⁴×C₆₇	1	48	yes	—
289	C₂²×C₄	C₂⁴×C₆₇	1	48	yes	—
290	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
291	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
292	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
293	C₄⋊C₄	C₂⁴×C₆₇	1	48	yes	—
294	C₄⋊C₄	C₂⁴×C₆₇	1	96	yes	—
295	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
296	C₂²⋊C₄	C₂⁴×C₆₇	1	32	yes	—
297	C₂²⋊C₄	C₂⁴×C₆₇	1	96	yes	—
298	C₄²	C₂⁴×C₆₇	1	48	yes	—
299	C₂³	—	2	32	no	—
300	C₂³	—	2	8	no	—
301	C₂³	—	2	4	no	—
302	C₂³	—	2	4	no	—
303	C₂³	—	2	4	no	—
304	C₂³	—	2	16	no	—
305	C₂³	—	2	8	no	—
306	C₂³	—	2	16	no	—
307	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
308	C₂³	2₋¹⁺⁴×C₆₇	1	8	yes	complemented
309	C₂³	C₂⁵×C₆₇	1	8	yes	—
310	C₂³	C₂⁵×C₆₇	1	2	yes	—
311	C₂³	C₂⁵×C₆₇	1	4	yes	—
312	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
313	Q₈	—	2	96	no	—
314	Q₈	—	2	48	no	—
315	Q₈	C₂²×D₄×C₆₇	1	64	yes	—
316	D₄	—	2	32	no	—
317	D₄	—	2	64	no	—
318	D₄	—	2	48	no	—
319	C₂×C₄	—	2	8	no	—
320	C₂×C₄	—	2	24	no	—
321	C₂×C₄	—	2	12	no	—
322	C₂×C₄	—	2	16	no	—
323	C₂×C₄	—	2	96	no	—
324	C₂×C₄	—	2	96	no	—
325	C₂×C₄	C₂²×D₄×C₆₇	1	24	yes	—
326	C₂×C₄	C₂²×D₄×C₆₇	1	48	yes	—
327	C₂×C₄	C₂⁵×C₆₇	1	16	yes	—
328	C₂×C₄	C₂⁵×C₆₇	1	12	yes	—
329	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
330	C₂×C₄	C₂⁵×C₆₇	1	12	yes	—
331	C₂²	—	2	32	no	—
332	C₂²	—	2	4	no	—
333	C₂²	—	2	4	no	—
334	C₂²	—	2	4	no	—
335	C₂²	—	2	8	no	—
336	C₂²	—	2	16	no	—
337	C₂²	—	2	8	no	—
338	C₂²	—	2	16	no	—
339	C₂²	—	4	16	no	—
340	C₂²	C₂³×D₄×C₆₇	1	4	yes	—
341	C₂²	C₂³×D₄×C₆₇	1	2	yes	—
342	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	4	yes	—
343	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	8	yes	complemented
344	C₂²	C₂⁶×C₆₇	1	1	yes	characteristic
345	C₄	—	2	8	no	—
346	C₄	—	2	12	no	—
347	C₄	—	2	16	no	—
348	C₄	C₂³×D₄×C₆₇	1	24	yes	—
349	C₂	—	2	4	no	—
350	C₂	—	2	8	no	—
351	C₂	—	4	8	no	—
352	C₂	C₆₇×#3486	1	4	yes	direct product factor
353	C₂	C₂⁴×D₄×C₆₇	1	1	yes	characteristic
354	C₂	C₂²×2₋¹⁺⁴×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₆₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×2₋¹⁺⁴×C₆₇	C₂	semidirect
2. C₂⁴×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#3486	C₂	direct (N × H)
4. C₂×D₄×Q₈×C₆₇	C₂	semidirect
5. C₂×#548×C₆₇	C₂	semidirect
6. C₂×#536×C₆₇	C₂	semidirect
7. C₂×#525×C₆₇	C₂	semidirect
8. C₂×#519×C₆₇	C₂	semidirect
10. C₂×2₋¹⁺⁴×C₆₇	C₂²	semidirect
12. C₂²×#41×C₆₇	C₂²	semidirect
17. C₂³×Q₈×C₆₇	C₂²	semidirect
18. C₂³×Q₈×C₆₇	C₂²	semidirect
20. D₄×Q₈×C₆₇	C₂²	semidirect
21. #548×C₆₇	C₂²	semidirect
22. #536×C₆₇	C₂²	semidirect
23. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
24. C₂×#124×C₆₇	C₂²	semidirect
25. #525×C₆₇	C₂²	semidirect
28. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
31. #519×C₆₇	C₂²	semidirect
32. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
40. C₂×#41×C₆₇	C₂³	semidirect
45. C₂²×Q₈×C₆₇	D₄	semidirect
52. C₂²×Q₈×C₆₇	C₂³	semidirect
62. C₄⋊Q₈×C₆₇	C₂³	semidirect
63. #124×C₆₇	C₂³	semidirect
66. C₂²⋊Q₈×C₆₇	C₂³	semidirect
69. C₄×Q₈×C₆₇	C₂³	semidirect
92. C₂×Q₈×C₆₇	C₂×D₄	semidirect
130. C₂³×C₆₇	2₋¹⁺⁴	semidirect
165. C₂²×C₆₇	C₂×2₋¹⁺⁴	semidirect
171. C₂×#3486	C₆₇	direct (N × H)
175. C₂×C₆₇	#3486	direct (N × H)
178. C₂²×2₋¹⁺⁴	C₂×C₆₇	semidirect
179. C₂⁴×Q₈	C₂×C₆₇	semidirect
180. #3486	C₂×C₆₇	direct (N × H)
181. C₂×D₄×Q₈	C₂×C₆₇	semidirect
182. C₂×#548	C₂×C₆₇	semidirect
183. C₂×#536	C₂×C₆₇	semidirect
184. C₂×#525	C₂×C₆₇	semidirect
185. C₂×#519	C₂×C₆₇	semidirect
186. C₆₇	C₂×#3486	direct (N × H)
188. C₂×2₋¹⁺⁴	C₂²×C₆₇	semidirect
190. C₂²×#41	C₂²×C₆₇	semidirect
195. C₂³×Q₈	C₂²×C₆₇	semidirect
196. C₂³×Q₈	C₂²×C₆₇	semidirect
198. D₄×Q₈	C₂²×C₆₇	semidirect
199. #548	C₂²×C₆₇	semidirect
200. #536	C₂²×C₆₇	semidirect
201. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
202. C₂×#124	C₂²×C₆₇	semidirect
203. #525	C₂²×C₆₇	semidirect
206. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
209. #519	C₂²×C₆₇	semidirect
210. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
218. C₂×#41	C₂³×C₆₇	semidirect
223. C₂²×Q₈	D₄×C₆₇	semidirect
230. C₂²×Q₈	C₂³×C₆₇	semidirect
240. C₄⋊Q₈	C₂³×C₆₇	semidirect
241. #124	C₂³×C₆₇	semidirect
244. C₂²⋊Q₈	C₂³×C₆₇	semidirect
247. C₄×Q₈	C₂³×C₆₇	semidirect
270. C₂×Q₈	C₂×D₄×C₆₇	semidirect
308. C₂³	2₋¹⁺⁴×C₆₇	semidirect
343. C₂²	C₂×2₋¹⁺⁴×C₆₇	semidirect
352. C₂	C₆₇×#3486	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×2₋¹⁺⁴×C₆₇	2
C₂⁴×Q₈×C₆₇	2
C₆₇×#3486	2
C₂×D₄×Q₈×C₆₇	2
C₂×#548×C₆₇	2
C₂×#536×C₆₇	2
C₂×#525×C₆₇	2
C₂×#519×C₆₇	2
C₂×#3486	67

Subgroups — C2×C67×#3486 #30742033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3486 #30742033